There wasn't a lecture either day; students did proofs at the board. I give some proofs similar to those given at the board. This is not the final version of these notes; I'm planning to add more proofs. all problems are from section 5.2. #3 A -> B | C |- (A -> B) | (A -> C) 1. A -> B | C premise Goal: (A -> B) | (A -> C) | 2. ~(A -> B) premise of subproof | Goal: A -> C | | 3. A premise of subsubproof | | Goal: C | | 4. B | C mp lines 1,3 | | Goal: ~B (so that disj. elim. can be applied to get C) | | | 5. B premise of subsubsubproof | | | Goal: contradiction | | | 6. A -> B true conclusion, line 5 (see a proof of this derived rule below) | | | 7. (A -> B) & ~(A -> B) c.i. lines 2,6 | | 8. ~ B neg. elim. lines 5-7 | | 9. C disj. elim. lines 4,8 | 10. A -> C lines 3-9 11. (A -> B) | (A -> C) disj. intro, lines 2-10. The rule of true conclusion used above is listed by Grantham. It has the form A ------- B -> A and can be proved valid as follows: 1. A (premise) Goal: B -> A | 2. B premise of subproof | 3. A copy of line 1 4. B -> A imp. intro, lines 2-3. The related rule of false antecedent given by Grantham can be proved thus: 1. ~A premise Goal: A -> B | 2. A subpremise | Goal: B | 3. B neg. elim. lines 2,1 4. A -> B imp. intro. lines 2-3 The rule can be presented thus: ~A ------------ A -> B #24 ~A | ~B |- ~(A & B) 1. ~A | ~B premise Goal: ~(A&B) | 2. A & B subpremise | Goal: contradiction | 3. A c.e. line 2 | 4. B c.e. line 2 | 5. ~~A d.n.i. line 3 | 6. ~B disj. elim. (disjunctive syllogism) lines 1,5 | 7. B & ~B c.i. lines 4,6 8. ~(A&B) neg. intro lines 2-7 #7 (A | B) | C |- A | (B | C) 1. (A | B) | C premise Goal: A | (B | C) | 2. ~A subpremise | Goal: B | C | | 3. ~C | | Goal: B | | 4. A | B disj. elim. lines 1,3 | | 5. B disj. elim. lines 2,4 | 6. B | C disj. intro. lines 3-5 7. A | (B | C) disj. intro. lines 2-6